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1
Pour 10 cups of water into a large soup pot over high heat with the leek, halved carrot, 3-inch piece of ginger, and peppercorns.
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2
Bring to a boil, turn heat to medium-low, and place chicken breasts into the broth
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3
Cover the pot with a lid and simmer the mixture until the chicken is cooked through and tender, 20 to 30 minutes.
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4
While broth is simmering, whisk flour and salt together in a bowl; beat in the eggs, butter, and enough milk to make a stiff but workable dough.
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5
Knead the dough in the bowl until smooth and slightly springy, about 5 minutes; transfer to a bowl, cover with a cloth, and allow to rest for 10 minutes.
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6
Roll the noodle dough out on a floured surface to desired thickness (1/4- to 1/8-inch thick); cut into strips.
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7
Remove chicken breasts, transfer them to a plate, and allow to cool. When cool, remove the bones and shred the meat with a fork.
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8
Strain and retain the broth; discard spent leek, halved carrot, ginger chunk, and peppercorns.
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9
Place strained broth into a large soup pot, bring to a boil, and reduce heat to medium-low. Stir in bouillon cubes, 3 tablespoons of chopped ginger, 3/4 cup sliced carrots, and celery until the bouillon has dissolved. Simmer until the carrots are tender, about 30 minutes.
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10
Gently stir in the noodles, a few at a time. Return broth to a simmer and cook until the noodles are tender, about 10 minutes.
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11
Whisk cornstarch and 1/4 cup of water until smooth; stir half the mixture into the soup and simmer until thickened, about 3 minutes. If thicker soup is desired, stir in remaining cornstarch mixture. Mix in the chicken to serve.
